Aggregating Experience Across Rating Periods

Why Multi-Period Aggregation Matters

An experience rating often uses data from several periods. Those periods may have different numbers of claims, different total losses, or different amounts of exposure. The rating indication depends on combining these pieces consistently. In particular, the experience measure should reflect the amount of risk represented by each period, rather than treating every period as equally informative by default.

This tutorial focuses on the arithmetic of combining periods: claim counts, losses, exposure, and period-level loss costs. The calculations assume that the periods are comparable in coverage, loss valuation basis, and dollar basis. No trend, development, or coverage adjustment is made here; if data require such adjustments, they must be addressed before aggregation. Exposure units also need a clear definition, such as earned policy-years or vehicle-years.

Definition: For period $j$, let $N_j$ be the number of claims, $L_j$ the total loss amount, and $E_j$ the exposure on a specified, consistent basis. The period claim frequency is $N_j/E_j$, the average claim severity is $L_j/N_j$ when $N_j>0$, and the period loss cost is $L_j/E_j$. Across $J$ periods, the pooled frequency, severity, and loss cost are respectively $\sum_j N_j/\sum_j E_j$, $\sum_j L_j/\sum_j N_j$ when the total claim count is positive, and $\sum_j L_j/\sum_j E_j$.

Pool Totals Before Calculating the Overall Loss Cost

The pooled loss cost is the total loss across all included periods divided by total exposure across those periods. It is a loss amount per exposure unit. It is not generally the unweighted arithmetic average of the period loss costs.

\text{Pooled loss cost} = \frac{\sum_{j=1}^{J} L_j}{\sum_{j=1}^{J} E_j}

If claim counts are available, the same pooled loss cost can be expressed as pooled frequency times pooled severity, provided the total claim count is positive:

\left(\frac{\sum_j N_j}{\sum_j E_j}\right) \left(\frac{\sum_j L_j}{\sum_j N_j}\right) = \frac{\sum_j L_j}{\sum_j E_j}

The identity follows because the total number of claims in the numerator of pooled frequency cancels the same total claim count in the denominator of pooled severity. This is a useful check on calculations. If the total claim count is zero, pooled severity is undefined, but pooled loss cost remains defined when total exposure is positive; in that case, total losses are also zero if losses are claim payments on the same basis.

The pooled loss cost can also be derived as an exposure-weighted average of period loss costs. Define $c_j=L_j/E_j$ for periods with positive exposure. Then $L_j=E_jc_j$, so:

\frac{\sum_j L_j}{\sum_j E_j} = \frac{\sum_j E_jc_j}{\sum_j E_j} = \sum_j\left(\frac{E_j}{\sum_k E_k}\right)c_j

The weights $E_j/\sum_k E_k$ sum to one. A period with more exposure therefore contributes more to the pooled loss cost. An ordinary arithmetic mean assigns equal weight to every period, which is appropriate only when the period exposures are equal or when equal period weighting is specifically intended.

Worked Example: Pooling Three Years of Claim Experience

A hypothetical liability program has the following experience. All losses are on the same valuation and dollar basis, and exposures are earned policy-years. Calculate the pooled frequency, severity, and loss cost. Compare the pooled loss cost with the unweighted average of the annual loss costs.

Period	Exposure	Claims	Total losses
1	100 policy-years	42	$210,000
2	120 policy-years	60	$330,000
3	80 policy-years	32	$160,000

Method. Add the counts, losses, and exposures separately, then form the pooled measures. This preserves the exposure basis and avoids giving the 80-policy-year period the same weight as the 120-policy-year period.

Step 1: Aggregate the data. Total exposure is $100+120+80=300$ policy-years. Total claims are $42+60+32=134$. Total losses are $\$210{,}000+\$330{,}000+\$160{,}000=\$700{,}000$.

Step 2: Calculate pooled frequency. The frequency is $134/300=0.446666\ldots$ claims per policy-year.

Step 3: Calculate pooled severity. Average severity is $\$700{,}000/134=\$5{,}223.880597\ldots$ per claim.

Step 4: Calculate pooled loss cost. The loss cost is $\$700{,}000/300=\$2{,}333.333\ldots$ per policy-year. As a check, pooled frequency times pooled severity is $0.446666\ldots\times \$5{,}223.880597\ldots=\$2{,}333.333\ldots$ per policy-year.

Step 5: Compare with period costs. The annual loss costs are $\$210{,}000/100=\$2{,}100$, $\$330{,}000/120=\$2{,}750$, and $\$160{,}000/80=\$2{,}000$. Their unweighted average is $(\$2{,}100+\$2{,}750+\$2{,}000)/3=\$2{,}283.33$. It differs from the pooled result because the annual exposures differ.

Check and common mistake. The exposure-weighted calculation is $(100\times \$2{,}100+120\times \$2{,}750+80\times \$2{,}000)/300=\$700{,}000/300$, matching the pooled loss cost. Using the unweighted average instead would understate this pooled result by $\$50$ per policy-year.

From Aggregated Experience to a Credibility Indication

Once the experience measure has been calculated on a consistent basis, it can be used in an experience-rated indication. For example, if a credibility weight $Z$ and a complement $C$ have already been supplied or selected, a standard credibility-weighted loss cost is $Z$ times the observed experience measure plus $1-Z$ times the complement. The aggregation step determines the observed experience input; it does not by itself determine $Z$.

\text{Indicated loss cost} = Z(\text{pooled experience loss cost})+(1-Z)C

The two terms must use the same units and basis. A pooled loss cost per policy-year cannot be combined directly with a complement stated per vehicle-year, for example. Likewise, if the supplied credibility formula calls for frequency or severity rather than loss cost, calculate and use the specified measure rather than substituting another one.

Worked Example: Credibility-Weighting Pooled Experience

A hypothetical equipment portfolio has two comparable periods. Period 1 has 50 exposure units and losses of $100,000; period 2 has 150 exposure units and losses of $390,000. Use a supplied credibility weight of $Z=0.40$ and a complement of $2,200 per exposure unit to calculate the indicated loss cost.

Method. First calculate the pooled loss cost; then apply the stated credibility formula. The credibility weight is given, so no credibility parameter estimation is required.

Step 1: Aggregate exposure and losses. Total exposure is $50+150=200$ units. Total losses are $\$100{,}000+\$390{,}000=\$490{,}000$.

Step 2: Calculate the pooled experience loss cost. It is $\$490{,}000/200=\$2{,}450$ per exposure unit.

Step 3: Apply credibility. The indicated loss cost is $0.40(\$2{,}450)+0.60(\$2{,}200)=\$980+\$1{,}320=\$2{,}300$ per exposure unit.

Check and common mistake. The period costs are $\$100{,}000/50=\$2{,}000$ and $\$390{,}000/150=\$2{,}600$. Their exposure-weighted mean is $(50\times \$2{,}000+150\times \$2{,}600)/200=\$2{,}450$. Credibility-weighting the unweighted period average of $2,300 would instead produce $0.40(\$2{,}300)+0.60(\$2{,}200)=\$2{,}240$, which is not the indicated result based on pooled experience.

Exposure Units Must Match the Risk Periods

A period’s exposure is not always equal to the number of contracts in force. If contracts have different durations, convert them to the exposure basis used in the rating calculation. For example, if the selected unit is policy-years, a six-month policy contributes one-half policy-year, assuming the exposure definition treats coverage uniformly over that term.

This conversion is not an unequal-exposure credibility formula. It is the basic accounting needed to express each period’s observed losses and claims per comparable unit. The exposure basis should be defined before calculating frequency or loss cost.

Worked Example: Converting Short-Term Policies to Policy-Years

A hypothetical specialty insurer observes 120 six-month contracts with 18 claims and $72,000 in losses, plus 30 annual contracts with 15 claims and $90,000 in losses. Treat every contract as fully earned for its stated term. Calculate pooled frequency, severity, and loss cost per policy-year.

Method. Convert each group’s contract count to policy-years before pooling. Then calculate the measures using the same steps as for periods already expressed in policy-years.

Step 1: Convert exposure. The 120 six-month contracts contribute $120\times 0.5=60$ policy-years. The 30 annual contracts contribute $30\times 1=30$ policy-years. Total exposure is $60+30=90$ policy-years.

Step 2: Aggregate claims and losses. Total claims are $18+15=33$. Total losses are $\$72{,}000+\$90{,}000=\$162{,}000$.

Step 3: Calculate pooled frequency. Frequency is $33/90=0.366666\ldots$ claims per policy-year.

Step 4: Calculate pooled severity and loss cost. Severity is $\$162{,}000/33=\$4{,}909.0909\ldots$ per claim. Loss cost is $\$162{,}000/90=\$1{,}800$ per policy-year. As a check, $0.366666\ldots\times \$4{,}909.0909\ldots=\$1{,}800$.

Check and common mistake. Counting all 150 contracts as 150 policy-years would give a loss cost of $\$162{,}000/150=\$1{,}080$ per supposed policy-year. That denominator is wrong: it treats each six-month contract as a full year. The correct result is $1,800 per policy-year.

Common Aggregation Traps

Averaging period loss costs without checking exposure. Use the pooled total loss divided by pooled exposure, or equivalently an exposure-weighted average of period costs, unless equal period weighting is explicitly required.
Averaging period severities when claim counts differ. Pooled severity is total losses divided by total claims. It is claim-count-weighted across periods, not generally an unweighted mean of period severities.
Mixing exposure units. Convert all periods to the same unit before calculating per-unit frequency or loss cost.
Combining inconsistent loss bases. Paid losses and incurred losses, or losses with different coverage terms, should not be combined as if they were directly comparable without the appropriate adjustments.
Rounding too early. Keep full precision through the aggregation and credibility calculation; round the final indicated amount to the requested precision.

An additional check is to reconstruct total losses: pooled loss cost multiplied by total exposure should equal the total losses used in the calculation. When the total claim count is positive, pooled frequency multiplied by pooled severity should also equal pooled loss cost. These checks can reveal unit errors, missing periods, and arithmetic mistakes.

Key takeaway: For a pooled experience measure, aggregate the underlying totals first: total claims, total losses, and total exposure. Divide only after confirming that the data use a consistent loss basis and exposure unit. This produces measures that reflect the amount of experience represented by each period.

Check Your Understanding

For each problem, use the stated exposure basis and retain full precision until the final answer.

A hypothetical program has losses of $96,000 on 80 exposure units in one period and $144,000 on 120 units in another. Calculate the pooled loss cost and compare it with the arithmetic average of the two period costs. (Answer: $1,200 per exposure unit; the arithmetic average is also $1,200 because the period exposures are equal in this case.)
Two periods have exposures of 40 and 80 policy-years, claim counts of 12 and 24, and losses of $48,000 and $144,000. Calculate pooled frequency, pooled severity, and pooled loss cost. (Answer: 0.3000 claims per policy-year; $5,333.33 per claim; $1,600 per policy-year.)
A portfolio has 25 exposure units with $50,000 of losses and 75 exposure units with $225,000 of losses. Calculate the two period loss costs and the pooled loss cost. (Answer: $2,000 and $3,000 per exposure unit; pooled loss cost $2,750 per exposure unit.)
Using the data in question 3, apply $Z=0.25$ and a complement of $2,200 per exposure unit. (Answer: $2,337.50 per exposure unit.)
A hypothetical book has 20 annual contracts with 10 claims and $50,000 of losses, and 80 fully earned three-month contracts with 14 claims and $70,000 of losses. Use policy-years as exposure. Calculate pooled frequency, severity, and loss cost. (Answer: 0.6000 claims per policy-year; $5,000 per claim; $3,000 per policy-year.)

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